Last updated Mon 1/31/11
The KU Combinatorics Seminar meets on Wednesdays, 3:00--4:00 PM in Snow 408. Please contact Jeremy Martin if you are interested in speaking.
Abstract: A tropical curve is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system |D| of a divisor D on a tropical curve analogously to the classical counterpart. We investigate the structure of |D| as a cell complex and show that linear systems are tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, |D| defines a map from the tropical curve to a tropical projective space, and the image can be extended to a parameterized tropical curve of degree equal to deg(D). The tropical convex hull of the image realizes the linear system |D| as an embedded polyhedral complex. We also show that curves for which the canonical divisor is not very ample are hyperelliptic. This is joint work with Christian Haase and Gregg Musiker.
Abstract: The Shi arrangement of hyperplanes plays an important role in the representation theory of affine Weyl groups. Its definition in type A is Shi(n)={x_i-x_j=0,1 : 1 ≤ i < j ≤ n}. This arrangement divides R^n into (n+1)^{n-1} regions --- an interesting number, yes? --- and it has beautiful combinatorics. In this talk I will introduce a new arrangement, called the Ish arrangement. You will like this arrangement. (Some of this is joint work with Brendon Rhoades.)
Last updated Mon 1/31/11